Mobile robots are enjoying increasing popularity in a number of different automation tasks. Omnidirectional mobile robots especially allow for a very flexible operation. They are able to accelerate in every direction, regardless of their orientation. In this context, we developed our own robot platform for research on said types of robots. It turns out that these mobile robots show interesting behaviour, which commonly used models for omnidirectional mobile robots fail to reproduce. As the exact sources and structures of mismatches are still unknown, non-parametric Gaussian process regression is used to develop a data-based model extension of the robot. A common control task for industrial applications is trajectory tracking, where a robot needs to follow a predefined path, for example in a warehouse, as close as possible in space and time. Appropriate feed-forward solutions for the data-based model are developed and finally leveraged in closed-loop control via nonlinear model predictive control. In real-world experiments, the results are compared to commonly used proportional position-based feedback. This novel contribution builds upon the preliminary work in [7] but, for the first time, includes also closed-loop (trajectory) tracking.